Beyond Shortest-Path Flow Fields: Certified Cycle-Exchange K-Flow Fields for Reusable Route Alternatives

Flow fields give agents sharing a destination constant-time action lookup, but conventional construction confines every route to shortest-path descent. Even when another corridor exists, agents cannot use it if all shortest routes converge at a bottleneck. We study complete destination-reaching flow fields as reusable policy families and prove their one-to-one correspondence with spanning trees, making every simple source-to-destination route realizable. We introduce Certified Cycle-Exchange \(K\)-Flow Fields (CCE-KFF), which compiles these families by exchanging tree edges. Each exchange preserves reachability from every source and identifies exactly which induced routes change, while the exchanges connect the complete field space. We also characterize the minimum number of exchanges between fields. Length- and topology-aware selectors produce near-shortest routes or alternatives through distinct corridors. Each compiled field remains executable from every reachable source. Experiments on Moving AI and StarCraft maps show stronger topological separation than source-specific \(K\)-shortest planners at near-shortest length. On Moving AI with 20 sources, CCE-KFF is over \(850\times\) faster than Yen while retaining constant-time action lookup.